Statistical inference on manifolds has seen significant attention in the modern literature. Many Frequentist approaches focus on problems where the data are on a manifold. [1]} develop asymptotic theory for mean and variance estimators using data on manifolds, with special attention given to the difference between intrinsic and extrinsic means, i.e. since the average of points on a sphere will exist off of the sphere, should one project this onto the sphere or develop a means to average directly on the sphere? In [2]}, a kernel density estimator is developed for the case where the sample space is a non-Euclidean manifold. In the Bayesian literature, posterior distributions are developed by first reparameterizing the problem to an unconstrained space [3]}, [4]}, or by taking a nonparametric approach [5]}, [6]}. [7]} discusses the invariance issues with Bayesian maximum a posteriori and minimum mean squared error estimates for the case where the Bayesian posterior is defined on a manifold. Many of these papers assume a priori that a manifold can be expressed by an explicitly known function, either by an isometric map from an abstract manifold to a Euclidean space, or by reparameterizing the problem to an unconstrained space.
